— DEV — Strain effects in freestanding nitride nanostructures

Attention

This tutorial is under construction

Input files:

• Strained-QW_AlGaN-GaN_Povolotskyi_PSS_2005_3D_nnp.in

Scope:

This tutorial aims to simulate strain effects in a lattice mismatched, freestanding heterostructure with wurtzite crystal structure consisting of an AlGaN/GaN quantum well. This tutorial is based on [Povolotskyi2005].

Output files:

• Strain/strain_simulation_2d_slice_middle_along_yz.vtr

• Strain/hydrostatic_strain_2d_slice_middle_along_yz.vtr

• Strain/strain_simulation_2d_slice_boundary_along_xz.vtr

• Strain/elastic_energy_density_2d_slice_middle_along_yz.vtr

Structure

Figure 2.4.149 shows the AlGaN/GaN/AlGaN quantum well structure, which is simulated in this tutorial. A 4 nm wide GaN QW layer is embedded between two \(\mathrm{Al_{0.28}Ga_{0.72}N}\) layers. The bottom AlGaN layer has a width of 10 nm, whereas the top AlGaN layer has a width of 6 nm. The overall shape of this nitride nanowire structure has the form of a cuboid with 50 nm x 50 nm extensions in the \(x\)- and \(y\)-directions. The height in the \(z\)-direction is 20 nm. The overall structure is surrounded by air (i.e. with a material where all elastic constants are zero).

The calculated strain pattern of this AlGaN/GaN structure is found to be highly non-homogeneous. The elastic energy has been minimized using continuum elasticity theory. We assume that the external stress applied to the structure is zero (freestanding structure).

Figure 2.4.149 Simulated heterostructure consisting of a GaN (green) layer sandwiched in between two AlGaN (blue) layers.

Results

Strain tensor components

In this section we show several strain tensor components \(\epsilon_{ij}(x, y, z)\) as a function of position \((x, y, z)\) for slices through the structure (vertical cross-section of the structure). Note that GaN has a larger lattice constant than AlGaN. Consequently, we expect the GaN layer to be compressively strained and the AlGaN layers to be tensely strained (or rather unstrained).

Figure 2.4.150 shows the strain tensor component \(\epsilon_{xx}\) along \(y\), \(z\) at \(x = 25.0\,\mathrm{nm}\). The corresponding data can be found in the file Strain/strain_simulation_2d_slice_middle_along_yz.vtr. The bottom AlGaN layer is rather unstrained (at the bottom), the GaN QW layer is strained compressively along the \(x\)-direction (blue region). This is not a surprise as we assumed coherent interfaces, i.e. the atomic planes match each other perfectly (pseudomorphic strain). The GaN QW induces a tensile strain to the AlGaN top layer (red region).

Figure 2.4.150 Calculated strain component \(\epsilon_{xx}\) in the \(y\), \(z\) plane at \(x = 25.0\,\mathrm{nm}\).

Figure 2.4.151 shows the strain tensor component \(\epsilon_{yy}\). Similar to Figure 2.4.150, the GaN layer is compressively strained (blue region), but only in the center and not at the boundaries, where it is nearly relaxed. Note that the \(\epsilon_{xx}\) and \(\epsilon_{yy}\) strain tensor components are not symmetric. This is due to the nitride crystal structure which has hexagonal symmetry perpendicular to the \((x, y)\) plane (and not cubic symmetry as the geometry of the structure).

• In the center (blue region), the GaN takes on the lattice constant of AlGaN (compressive strain).

• At the QW boundaries, the GaN takes on the lattice constant of ~GaN (nearly fully relaxed).

• Below and above the QW boundaries (red regions), the AlGaN takes on the lattice constant of ~GaN (tensile strain).

Figure 2.4.151 Calculated strain component \(\epsilon_{yy}\) in the \(y\), \(z\) plane at \(x = 25.0\,\mathrm{nm}\).

Figure 2.4.152 shows the strain tensor component \(\epsilon_{zz}\). As the GaN layer is compressively strained along both the \(x\)- and \(y\)-directions, it is tensilely strained (green region) along the \(z\)-direction (biaxially strained GaN layer, Poisson ratio).

Figure 2.4.152 Calculated strain component \(\epsilon_{zz}\) in the \(y\), \(z\) plane at \(x = 25.0\,\mathrm{nm}\).

Figure 2.4.153 shows the hydrostatic strain \(\epsilon_\mathrm{hydro} = \epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz}\), which is the trace of the strain tensor, i.e. the sum of the diagonal strain tensor components. It corresponds to the overall volume change. The data can be found in the file Strain/hydrostatic_strain_2d_slice_middle_along_yz.vtr. The blue region indicates that the GaN is strained compressively. AlGaN is mostly unstrained apart from the red regions at the left and right boundaries of the material interfaces. In a real sample, due to the deformation, the heterostructure changes its shape and becomes bent. In our case, the strain is small (less than \(1\%\)), so the shape of the structure does not change significantly.

Figure 2.4.153 Calculated hydrostatic strain component \(\epsilon_\mathrm{hydro} = \epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz}\) in the \(y\), \(z\) plane at \(x = 25.0\,\mathrm{nm}\).

In contrast to heterostructures, which are infinitely large and homogeneous in the lateral directions (i.e. in the \((x,y)\) plane), the deformation of a structure of a finite size is not homogeneous, as e.g. in GaN nanowire heterostructures. Since the structure is grown along the high symmetry direction [0001], the strain patterns exhibits reflection symmetry along the axis through the center (oriented parallel to the \(z\)-axis). The deformation becomes homogeneous in the region near the central axis, reproducing the case of an infinitely large structure.

Figure 2.4.154 shows the off-diagonal strain tensor component \(\epsilon_{yz}\). The strain tensor components \(\epsilon_{xy}\) and \(\epsilon_{xz}\) are zero for this particular slice. (In fact, the maximum value of \(\epsilon_{xy}\) is an order of magnitude smaller that the maximum value of \(\epsilon_{xz}\) or \(\epsilon_{yz}\).)

Figure 2.4.154 Calculated strain component \(\epsilon_{yz}\) in the \(y\), \(z\) plane at \(x = 25.0\,\mathrm{nm}\).

Figure 2.4.155 shows the same off-diagonal strain tensor component \(\epsilon_{yz}\), but this time at slices at the left and right boundaries. The corresponding data can be found in the file Strain/strain_simulation_2d_slice_boundary_along_xz.vtr

Figure 2.4.155 Calculated strain component \(\epsilon_{yz}\) in the \(x\), \(z\) plane at the boundaries.

Elastic energy density

Due to a possible usage of such structures as a light emitter, the strain in the GaN layer where charge carriers are confined, is or particular interest, i.e. the influence of stain on the conduction and valence band structure through deformation potentials. Additional, piezoelectric and pyroelectric fields have to be taken into account. The piezoelectric fields depend on the strain distribution in the sample. Thus, both the piezoelectric field and the GaN energy gap will vary along the lateral direction.

Figure 2.4.156 shows the energy density of the elastic deformation in units of \([\mathrm{eV/nm^3}]\). The corresponding data can be found in the file Strain/elastic_energy_density_2d_slice_middle_along_yz.vtr. The accumulated elastic energy in the pseudomorphically grown GaN QW is gradually reduced towards the free surface along the lateral direction. Consequently, the GaN QW center is almost fully strained, whereas towards the lateral surface there is a continuous relaxation.

Figure 2.4.156 Elastic energy density in the \(x\), \(z\) plane at \(x = 25.0\,\mathrm{nm}\).

Last update: 17/07/2024